In this contribution, we focus solely on the one-to-one correspondence
of solutions of flat systems to solutions of a trivial system with
the same number of inputs, as it is known from the continuous-time
case. For discrete-time systems, this would mean that the flat output
may depend both on forward- and backward-shifts of the system variables.
Conversely, the state- and input variables could be expressed as functions
of both forward- and backward-shifts of the flat output. To distinguish
the usual definition of [1]}, [2]}
and [3]} with forward-shifts
from the alternative approach of the present paper, we refer to the
former as forward-flatness. A special case of this alternative definition
has already been suggested in [4]}, where
the flat output may depend also on backward-shifts of the input variables
but not on backward-shifts of the state variables.For systems where the Jacobian matrix of the right-hand side of the
system equations with respect to the state variables has full rank,
both approaches are equivalent. To justify our alternative approach, we show that flat systems (in
the extended sense with backward-shifts) still share many beneficial
properties of forward-flat systems. In particular, we show that they
still are reachable (and hence controllable) and allow a straightforward
planning of trajectories to connect arbitrary points of the state
space. Furthermore, we show that they can be linearized by a dynamic
feedback. With respect to the classical dynamic feedback linearization
problem, the following inclusions hold: (static feedback linearizable
systems) \(\subset \)  (forward-flat systems) \(\subset \)  (flat systems)
\(\subset \)  (dynamic feedback linearizable systems). We illustrate
by examples that the class of forward-flat systems is a strict subset
of the class of flat systems in the alternative sense with backward-shifts.
